NPTEL lectures on Classical Field Theory

This is my page for my NPTEL video course that was recorded during my Fall 2010 lectures to a live class. You are expected to solve the problem sets as suggested below. No solutions are provided as they are almost always discussed during the later lectures. So it is better to try to solve the problem sets even if you can't get the correct solution.

A request: If you find errors in any of the lectures, send an
email to suresh.govindarajan at gmail.com pointing out the error along with
the lecture number and time on the video. Please accept my apologies for the errors which are (hopefully) of a minor nature.

Credits: The table of contents for the lectures have been provided by Naveen S. P., S. Sivaramakrishnan and T. Srinidhi who attended the live lectures. The following have pointed out errors in the lectures: Sayani Chatterjee, S. Sivaramakrishnan, Arun K. J.

I have put up twelve assignments, one quiz and one final examination (as pdf files). It is not optional but a MUST that you do the assignments as indicated below.1 Other supporting material (handouts, links) will be added soon! Please be patient.

### Module 1: Introduction to Classical Field Theory (1 Lecture)

Date Lecture Number Content of the Lecture Additional Info
Aug. 05, 2010 Lecture 1: What is Classical Field Theory? Review of classical mechanics, Particle Trajectories and the Principle of least action, Feynman's description of QM, Classical Mechanics to Classical Fields. Do Problem Set 1 before viewing Lecture 2!

### Module 2: Symmetries and Group Theory (6 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Aug. 09, 2010 Lecture 2: Symmetries and Invariances - I Symmetries, Invariances of Newton's EOM vs Maxwell's Equations, The Galilean Group.
Aug. 12, 2010 Lecture 3: Symmetries and Invariances - II Invariances of Maxwell's Equations continued, Common Four Vectors, Covariant Formulation of Maxwell's Equations,Lorentz and Poincare Groups, Rotation Group and vectors under rotation. Attempt Problem Set 2 after viewing Lecs. 2 and 3.
Aug. 13, 2010 Lecture 4: Group Theory in Physics - I Definition of a Group, Antisymmetric Matrices and $SO(d)$,Vectors and Tensors of $SO(d)$, Parity: Polar and Axial Vectors. Solve Problem Set 3 while/after viewing lecs. 4 and 5.
Aug. 16, 2010 Lecture 5 Group Theory in Physics - II Generalizations of $SO(d)$ (specifically the Lorentz Group),Simple Boost Matrices and Rapidity, $SO(p,q)$ with general signatures in metric,The Symplectic Group. 40:30 The matrix should be symmetric and non-degenerate. Symplectic matrices have det=1
Aug. 16, 2010 Lecture 6: Finite Groups - I Finite Groups of low order : Cyclic and Coxeter(specifically Dihedral) Groups,Definition of a Subgroup, Equivalence relation and Cosets. 49:19 Left coset wrongly called right coset. Corrected at the start of lec. 7
Aug. 19, 2010 Lecture 7: Finite Groups - II Left and Right Cosets, Permutation Group,Normal Subgroups, Classification of Finite Simple Groups, Monstrous moonshine. Solve Problem Set 4 while/after viewing lecs. 6 and 7. Normal Subgroups

My colleague, Prof. Balakrishnan has given three lectures(Lec 1) on group theory as a part of his NPTEL Mathematical Physics course. You can use these lectures as a supplement to my lectures if you wish.

### Module 3 Actions for Classical Field Theory (3 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Aug. 20, 2010 Lecture 8: Basics of CFT - I Classical Mechanics of Fields, Structure of the KE term in the Lagrangian density, the ultra-local term, and Lorentz invariance of the Lagrangian. Solve Problem Set 5 after viewing lecs. 8 and 9.
Aug. 26, 2010 Lecture 9: Basics of CFT - II Action Principle for fields, Conditions on Lagrangian density for no surface contribution, Conserved Currents,Hamiltonian density, Conditions for Finite Energy. at 50:46 and 51:33 min[finite energy cond — wrong power]
Aug. 27, 2010 Lecture 10: Basics of CFT - III Definition of Vacuum and examples, Vacuum Solutions for quartic potential, Topological Currents and Charges, Noether's Theorem, Application to translational invariance. Solve Problem Set 6 after viewing lec. 10 but before lec. 15 where it is discussed.

### Module 4 Green Functions for the Klein-Gordon Operator (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Aug. 30, 2010 Lecture 11 Green Functions - I Inhomogenous Klein-Gordon Equation, Method of the Green functions, Advanced and Retarded Green Functions. Handout on Green Functions
Aug. 30, 2010 Lecture 12 Green Functions - II Green Functions of the KG operator, Closing the contour,The Feynman propagator. There are some errors. See handout.

My colleague, Prof. Balakrishnan discusses the Green function for several families of operators as a part of his NPTEL Mathematical Physics course. You can use these lectures as a supplement to my lectures if you wish.

### Module 5 Symmetries and Conserved quantities (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Sept. 09, 2010 Lecture 13 Noether's Theorem - I Types of Symmetries, Internal Symmetries, Notion of "small", Transformations to first order (for Lorentz Group), Formulation to derive the Master formula. 21:00 Cleaner proof of antisymmetry of infinitesimal Lorentz transformations
Sept. 06, 2010 Lecture 14 Noether's Theorem - II Derivation of the Master formula for the Noether current, The energy-momentum tensor and the generalized angular momentum tensor as examples. Solve Problem Set 7 after viewing lec. 14 but before lec. 20 where it is discussed.

### Module 6 Solitons - I (Kink soliton) (1 lecture)

Date Lecture Number Content of the Lecture Additional Info
Sept. 06, 2010 Lecture 15 Kink Soliton Studying time-independent, finite energy solutions to the Euler-Lagrange equations of motion,the kink soliton, Derrick's theorem and its proof.

### Module 7 Hidden Symmetry (Spontaneous Symmetry Breaking) & the abelian Higgs mechanism (3 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Sept. 09, 2010 Lecture 16: Hidden Symmetry Spontaneous symmetry breaking and statement of Goldstone's theorem.
Sept. 13, 2010 Lecture 17: Local Symmetries Symmetry breaking continued, The Mermin-Wagner-Coleman theorem,The ideas of global and local symmetries, the covariant derivative,Minimal prescription for the covariant derivative. 23:13 Should change the location of the minus sign in the $SO(2)$ matrix or equivalently take $q$ to $-q$.
Sept. 13, 2010 Lecture 18 The Abelian Higgs model Definition of field strength using the covariant derivative, Small fluctuations about the vacuum solution,The Higgs mechanism in the $U(1)$ case 29:25: Index mismatch in cov. current $\mu$/$\nu$ on LHS/RHS.

### Module 8 Lie algebras, symmetry breaking and Noether's theorem for Maxwell Equations (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Sept. 16, 2010 Lecture 19: Lie Algebras - I Recap of symmetries and Noether's theorem, Lie algebras and finite-dimensional representations, $su(2)$ Lie Algebra. Solve Problem Set 8 while viewing lectures 19/20.
Sept. 17, 2010 Lecture 20 Lie Algebras - II $su(3)$ Lie Algebra ; Symmetry breaking in terms of Lie algebras, Conserved currents for the Proca action: energy-momentum, generalized angular momentum and the symmetric energy-momentum tensors. Equivalences of Lie algebras

### Module 9 Solitons — II (Magnetic Vortices) (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Sept. 27, 2010 Lecture 21: Magnetic Vortices - I Finite energy, time-independent solutions in the Abelian Higgs model in 2+1 dimensions, Topological charge == Magnetic flux, Quantization of magnetic flux, The Bogomol'nyi-Prasad-Sommerfield(BPS) bound for energy, Saturation of the BPS bound. Solve Problem Set 9 before viewing lecture 30.
Oct. 04, 2010 Lecture 22: Magnetic Vortices - II Vortices in the Abelian Higgs model applied to superconducting materials,characteristic lengths in the problem, "size" of a vortex, Description of vortex number using the fundamental group of the gauge group $U(1)$, or the circle. 35:38 and 36:03 min[finite energy cond — wrong power]

### Module 10 Towards Non-abelian gauge theories (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Oct. 07, 2010 Lecture 23: Non-abelian gauge theories - I Non-abelian gauge symmetry with SU(2) as an example, Covariant derivative in the non-abelian case, Construction of a locally SU(2) invariant Lagrangian, Transformation of the gauge fields under local gauge transformations. Solve Problem Set 10 while viewing lectures 23/24.
Oct. 08, 2010 Lecture 24: Non-abelian gauge theories - II Transformation of the gauge fields(continued), Derivation of the field strength for the gauge field, Symmetry breaking in the non-abelian case, Goldstone's theorem in terms of Lie Algebras.

### Module 11 Representation theory of Lie Algebras (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Oct. 11, 2010 Lecture 25: Irreps of Lie algebras - I Representation theory of $su(2)$ and $su(3)$, the Cartan subalgebra, the adjoint representation. 3:00 - Misleading statement: Map from G to GL(N). While GL(N) is set of all linear maps on V, the map from G to GL(N) is not linear.28:00 - Blocks "0" and "*" in the matrix have been interchanged.
Oct. 14, 2010 Lecture 26: Irreps of Lie algebras - II Representation theory continued, Ferrer's diagrams.

### Quiz (Test yourself)

If you have gotten this far, you can test you understanding by taking this Quiz. It is meant to be an open notes (i.e., your own notes) examination and the expected duration is one and half hours. I don't intend to post the solutions online but they will be provided on request. This is to counter the natural human tendency to look at solutions if they are available! What is a good score? I would say anything over 50% is acceptable.

### Module 12 The Standard Model of Particle Physics (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Oct. 15, 2010 Lecture 27 The Standard Model - I $su(3)$ multiplets, Motivation for the Standard Model, Colour confinement, Gell-Mann—Nishijima relation.
Oct. 18, 2010 Lecture 28 The Standard Model - II Electroweak symmetry breaking — An application of symmetry breaking in the non-abelian case. Typo towards the end: The prediction is that there is a massive (not massless) particle.

### Module 13 The Lorentz and Poincare Lie Algebras (1 Lecture)

Date Lecture Number Content of the Lecture Additional Info
Oct. 21, 2010 Lecture 29: Irreps of the Lorentz/Poincare algebras The Lorentz and Poincare algebras and their representations.

### Module 14 Solitons — III (Monopoles and Dyons) (3 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Oct. 22, 2010 Lecture 30: The Dirac mononpole Magnetically charged solutions: The Dirac monopole, Flux quantization. Solve Problem Set 11 while viewing lectures 30/31/32.
Oct. 25, 2010 Lecture 31: The 't Hooft-Polaykov monopole Magnetically charged solutions: The ‘t Hooft-Polyakov monopole, The Prasad-Sommerfield limit.
Oct. 28, 2010 Lecture 32: Revisiting Derrick’s Theorem Revisiting Derrick's theorem, BPS solution
Nov. 1, 2010 Lecture 33: The Julia-Zee dyon Constructing dyonic solutions, Dirac quantization for dyons; Dimensional reduction.

### Module 15 Instantons and their physical interpretation (4 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Nov. 4, 2010 Lecture 34: Instantons - I Quantum mechanical tunnelling and Instantons. Solve Problem Set 12 while viewing lectures 34/37.
Nov. 8, 2010 Lecture 35:| Instantons - II Kink soliton and tunnelling, Instantons in pure Yang-Mills theories(SU(2)).
Nov. 11, 2010 Lecture 36: - Instantons - III More on instantons, The BPS bound.
Nov. 12, 2010 Lecture 37: Instantons - IV Free parameters in instanton solutions, moduli space, Complexified Yang-Mills and theta vacua.

### Module 16 An introduction to some advanced topics (2 Lectures)

Date Lecture Number Content of the Lecture Additional Info
Nov. 15, 2010 Lecture 38: Dualities Dualities in Field Theory: Ising Model; Sine-Gordon / Massive Thirring; SU(2) Yang-Mills in 3+1 dimensions.
Nov. 18, 2010 Lecture 39: Geometrization of Field Theory General relativity as a gauge theory; Geometrization of Field Theory; Glimpse into String theory and branes.

### The Final

If you have gotten this far, you can test your understanding (of the course material) by taking this Final Examination. It is meant to be an open notes (i.e., your own notes) examination and the expected duration is three hours. I don't intend to post the solutions online but they will be provided on request.